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Flat knot 6.82

Min(phi) over symmetries of the knot is: [-4,-2,0,1,1,4,0,2,2,4,4,1,1,2,2,1,1,3,0,1,3]
Flat knots (up to 7 crossings) with same phi are :['6.82']
Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 10*K1**2 - 8*K1*K2 + K1 - 4*K2**2 + 3*K2 + 3*K3 + K4 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.82']
Outer characteristic polynomial of the knot is: t^7+109t^5+60t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.82']
2-strand cable arrow polynomial of the knot is: -64*K1**6 + 928*K1**4*K2 - 3808*K1**4 + 832*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1056*K1**3*K3 - 128*K1**2*K2**4 + 480*K1**2*K2**3 + 160*K1**2*K2**2*K4 - 5472*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 64*K1**2*K2*K4**2 - 1184*K1**2*K2*K4 + 10384*K1**2*K2 - 1680*K1**2*K3**2 - 64*K1**2*K3*K5 - 608*K1**2*K4**2 - 32*K1**2*K4*K6 - 7560*K1**2 + 544*K1*K2**3*K3 + 64*K1*K2**2*K3*K4 - 992*K1*K2**2*K3 - 96*K1*K2**2*K5 + 160*K1*K2*K3**3 + 64*K1*K2*K3*K4**2 - 448*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 9168*K1*K2*K3 - 64*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 3856*K1*K3*K4 + 832*K1*K4*K5 + 32*K1*K5*K6 + 8*K1*K6*K7 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 160*K2**4*K4 - 920*K2**4 + 32*K2**3*K3*K5 + 64*K2**2*K3**2*K4 - 1008*K2**2*K3**2 + 32*K2**2*K4**3 - 416*K2**2*K4**2 + 1656*K2**2*K4 - 5582*K2**2 - 160*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 760*K2*K3*K5 + 344*K2*K4*K6 - 240*K3**4 - 208*K3**2*K4**2 + 208*K3**2*K6 - 3508*K3**2 + 160*K3*K4*K7 - 48*K4**4 + 40*K4**2*K8 - 1790*K4**2 - 356*K5**2 - 106*K6**2 - 32*K7**2 - 10*K8**2 + 6702
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.82']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16943', 'vk6.17185', 'vk6.20556', 'vk6.21955', 'vk6.23340', 'vk6.23634', 'vk6.28010', 'vk6.29475', 'vk6.35381', 'vk6.35800', 'vk6.39418', 'vk6.41609', 'vk6.42854', 'vk6.43131', 'vk6.45994', 'vk6.47668', 'vk6.55094', 'vk6.55347', 'vk6.57424', 'vk6.58593', 'vk6.59493', 'vk6.59784', 'vk6.62091', 'vk6.63067', 'vk6.64937', 'vk6.65144', 'vk6.66960', 'vk6.67819', 'vk6.68227', 'vk6.68369', 'vk6.69571', 'vk6.70266']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is c.
The reverse -K is
The mirror image K* is
The reversed mirror image -K* is
The fillings (up to the first 10) associated to the algebraic genus:
Or click here to check the fillings

invariant value
Gauss code O1O2O3O4O5O6U2U4U6U3U1U5
R3 orbit {'O1O2O3O4O5O6U2U4U6U3U1U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U2U6U4U1U3U5
Gauss code of K* O1O2O3O4O5O6U5U1U4U2U6U3
Gauss code of -K* O1O2O3O4O5O6U4U1U5U3U6U2
Diagrammatic symmetry type c
Flat genus of the diagram 3
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -1 -4 0 -1 4 2],[ 1 0 -3 1 0 4 2],[ 4 3 0 3 1 4 2],[ 0 -1 -3 0 -1 2 1],[ 1 0 -1 1 0 2 1],[-4 -4 -4 -2 -2 0 0],[-2 -2 -2 -1 -1 0 0]]
Primitive based matrix [[ 0 4 2 0 -1 -1 -4],[-4 0 0 -2 -2 -4 -4],[-2 0 0 -1 -1 -2 -2],[ 0 2 1 0 -1 -1 -3],[ 1 2 1 1 0 0 -1],[ 1 4 2 1 0 0 -3],[ 4 4 2 3 1 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-2,0,1,1,4,0,2,2,4,4,1,1,2,2,1,1,3,0,1,3]
Phi over symmetry [-4,-2,0,1,1,4,0,2,2,4,4,1,1,2,2,1,1,3,0,1,3]
Phi of -K [-4,-1,-1,0,2,4,0,2,1,4,4,0,0,1,1,0,2,3,1,2,2]
Phi of K* [-4,-2,0,1,1,4,2,2,1,3,4,1,1,2,4,0,0,1,0,0,2]
Phi of -K* [-4,-1,-1,0,2,4,1,3,3,2,4,0,1,1,2,1,2,4,1,2,0]
Symmetry type of based matrix c
u-polynomial -t^2+2t
Normalized Jones-Krushkal polynomial 3z^2+24z+37
Enhanced Jones-Krushkal polynomial 3w^3z^2+24w^2z+37w
Inner characteristic polynomial t^6+71t^4+19t^2
Outer characteristic polynomial t^7+109t^5+60t^3+5t
Flat arrow polynomial 4*K1**3 + 4*K1**2*K2 - 10*K1**2 - 8*K1*K2 + K1 - 4*K2**2 + 3*K2 + 3*K3 + K4 + 7
2-strand cable arrow polynomial -64*K1**6 + 928*K1**4*K2 - 3808*K1**4 + 832*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1056*K1**3*K3 - 128*K1**2*K2**4 + 480*K1**2*K2**3 + 160*K1**2*K2**2*K4 - 5472*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 64*K1**2*K2*K4**2 - 1184*K1**2*K2*K4 + 10384*K1**2*K2 - 1680*K1**2*K3**2 - 64*K1**2*K3*K5 - 608*K1**2*K4**2 - 32*K1**2*K4*K6 - 7560*K1**2 + 544*K1*K2**3*K3 + 64*K1*K2**2*K3*K4 - 992*K1*K2**2*K3 - 96*K1*K2**2*K5 + 160*K1*K2*K3**3 + 64*K1*K2*K3*K4**2 - 448*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 9168*K1*K2*K3 - 64*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 3856*K1*K3*K4 + 832*K1*K4*K5 + 32*K1*K5*K6 + 8*K1*K6*K7 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 160*K2**4*K4 - 920*K2**4 + 32*K2**3*K3*K5 + 64*K2**2*K3**2*K4 - 1008*K2**2*K3**2 + 32*K2**2*K4**3 - 416*K2**2*K4**2 + 1656*K2**2*K4 - 5582*K2**2 - 160*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 760*K2*K3*K5 + 344*K2*K4*K6 - 240*K3**4 - 208*K3**2*K4**2 + 208*K3**2*K6 - 3508*K3**2 + 160*K3*K4*K7 - 48*K4**4 + 40*K4**2*K8 - 1790*K4**2 - 356*K5**2 - 106*K6**2 - 32*K7**2 - 10*K8**2 + 6702
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {3, 5}, {1, 4}]]
If K is slice False
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